Simulating transport and distribution of marine macro-plastic in the Baltic Sea

We simulated the spatial distribution and dynamics of macro plastic in the Baltic Sea, using a new Lagrangian approach called the dynamical renormalization resampling scheme (DRRS). This approach extends the super-individual simulation technique, so the weight-per-individual is dynamic rather than fixed. The simulations were based on a mapping of the macro plastic sources along the Baltic coast line, and a five year time series of realistic wind, wave and current data to resolve time-variability in the transport and spatial distribution of macro plastics in the Baltic Sea. The model setup has been validated against beach litter observations and was able to reproduce some major spatial trends in macroplastic distributions. We also simulated plastic dispersal using Green’s functions (pollution plumes) for individual sources. e.g. rivers, and found a significant variation in the spatial range of Green’s functions corresponding to different pollution sources. We determined a significant temporal variability (up to 7 times the average) in the plastic concentration locally, which needs to be taken into account when assessing the ecological impact of marine litter. Accumulation patterns and litter wave formation were observed to be driven by an interplay between positive buoyancy, coastal boundaries and varying directions of physical forcing. Finally we determined the range of wind drag coefficients for floating plastic, where the dynamics is mostly directly wind driven, as opposed to indirectly by surface currents and waves. This study suggests that patterns of litter sorting by transport processes should be observable in many coastal and off-shore environments.

The first simple reference system is the 1D case in the no advection limit v = 0, so x is scalar. We arbitrarily place the source with an influx of S units per time at x = 0. At stationarity we get which is solved by G = A exp(−kx) with k ± = ± λ/D h and B = S/ √ λD h , if the problem is confined to x > 0. If free boundary conditions apply to the right, only the k + branch applies.
Second we consider downstream advection v > 0 and get and we again find exponential solutions with for the downstream side (x > 0), giving again k ± = λ/D h in the diffusive limit (v → 0), and k = λ v in the advective limit (D h → 0). . If free boundary conditions apply to the right, only the k + branch applies in the diffusive limit. If the problem is confined to x > 0, this implies B = S/(v + D h k + ). If the problem is solved with specific right side boundary conditions at x = L, e.g. G(L) = P 0 , the solution is a linear combination of branches k ± satisfying flux boundary conditions to the left x = 0 and the specified boundary condition to the right (x = L). The limiting solutions suggest to consider the dimensionless number which tells whether the pollution plume tail is advection like (χ >> 1) or diffusion like (χ << 1). In this way, χ corresponds to the Péclet number for the advection-diffusion problem, telling whether transport on a given scale is advection or diffusion like. An alternative way to identify that χ characterizes the dynamical regime is that χ and 1/χ gives the leading correction to k in the diffusion and advection limit, respectively.
Exploring dimensionality, we may consider the 2D diffusion-dissipation problem at stationarity with the source at r = 0: where the regular solution is again with k = λ/D h as for 1D, and being the modified Bessel's function of second kind and 0 th order.
S2 Back diffusion correction In the derivation of the DRRS scheme, we included only advective contributions along the open (water bound) boundaries of the system. If advection locally were into the system that was handled as a local plastic point source.
If the Lagrangian ratio is small, diffusive transport is locally dominating, for a given particle time step dt. This means that a small number of particles will diffuse back into the system, if the boundary concentration P 0 > 0, even if the advective direction is out of the system, for small ξ. For the 1D case, it is follows from Eq. 3 (using normal variates for η for analytical convenience) that the number of back diffusing particles in a time step dt is If P 0 >> 0 and ξ << 1 point sources along the open boundaries corresponding to may be applied in the DRRS scheme. Backdiffusing particles should be distributed according away from the boundary normal, with z = 0 corresponding to the boundary, and z < 0 the system interior, if Gaussian statistics is applied for η. Sampling from distribution Eq. 25 is not commonly supported in numerical libraries, but n variates from Eq. 25 are efficiently generated by solving C(U (0, 1, n)) = x) with a few vectorized Newton-Raphson steps, where C is the cumulative density of Eq. 25, and U (0, 1, n) a random variate of length n on ]0, 1]. Corresponding equations can be derived for 1D/2D boundaries, corresponding to positively buoyant particles or microplastic dispersed vertically. From a technical perspective, neglecting the Lagrangian back diffusion correction strictly corresponds to solving the Eulerian problem Eq. 15 with boundary condition P 0 = 0 to the right side. The back diffusion correction, when P 0 >> 0 at the boundary is usually small and may be neglected in many real cases; in our paper case study we did not include the back diffusion correction, corresponding to assuming  oscillations around the correct asymptotic value (green line). Figure 11(b) show the convergence of the standard deviation ∆ M T = < (G M T (x) − G(x)) 2 > for both algorithms (T not applicable for non-resampling scheme, where R is fixed); this plot shows that the standard deviation ∆ M T converges much faster for the DRRS algorithm, as heralded previously, so that the same accuracy can be achieved with fewer particles in the simulation (i.e. the simulation is computationally faster), or a more accurate result can be achieved with the same calculational load. The reason for the efficiency gain is that all particles in the simulation are active throughout the simulation in DRRS, whereas some must be reserved in the non-resampling algorithms so that continuous release of plastic at x = 0 can be simulated (in our benchmark all particles in the non-resampling Lagrangian run were active at some point in the simulation).  Actual physical forcing databases not distributed with the IBMlib framework.

S4.2 1D DRRS demonstration sandbox
The Lagrangian simulation demonstration of the simplest spatial reference system of a pollution plume analyzed in S1 Spatial scaling of Green's functions, S2 Back diffusion correction and S3 Benchmarking the DRRS algorithm. The object-oriented demonstration is implemented in Python3, using NumPy and SciPy, and dynamics vectorized for optimal performance.
This includes scripts that setup baseline simulations (Greens functions and regional plastic distribution dynamics) with parameterization as described in this paper. This also includes the plastic source map input file and a README file explaining the setup and how to obtain physical forcing data. The installation and requirements of IBMlib are described in https://github.com/IBMlib/IBMlib/tree/master/doc .